A156741 - OEIS

%I #13 Sep 04 2024 18:58:16

%S 1,1,1,1,190,1,1,7315,7315,1,1,134596,5181946,134596,1,1,1562275,

%T 1106715610,1106715610,1562275,1,1,13123110,107904771975,

%U 1985447804340,107904771975,13123110,1,1,86493225,5974000557525,1275875833357125,1275875833357125,5974000557525,86493225,1

%N Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 8, read by rows.

%H G. C. Greubel, <a href="/A156741/b156741.txt">Rows n = 0..30 of the triangle, flattened</a>

%F T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 8.

%F Sum_{k=0..n} T(n, k) = A151709(n).

%e Triangle begins as:

%e   1;

%e   1,        1;

%e   1,      190,            1;

%e   1,     7315,         7315,             1;

%e   1,   134596,      5181946,        134596,            1;

%e   1,  1562275,   1106715610,    1106715610,      1562275,        1;

%e   1, 13123110, 107904771975, 1985447804340, 107904771975, 13123110, 1;

%t b[n_, k_]:= Binomial[2*n, 2*k];

%t T[n_, k_, m_]:= Round[Product[b[n+j, k+j]/b[n-k+j, j], {j,0,m}]];

%t Table[T[n, k, 8], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 19 2021 *)

%o (Magma)

%o A156741:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..8]]) ) >;

%o [A156741(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 19 2021

%o (Sage)

%o def A156741(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..8)) )

%o flatten([[A156741(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 19 2021

%Y Cf. A086645 (m=0), A156739 (m=6), A156740 (m=7), this sequence (m=8), A156742 (m=9).

%Y Cf. A151709 (row sums).

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 14 2009

%E Definition corrected to give integral terms and edited by _G. C. Greubel_, Jun 19 2021